The general solution for the vibrational problem is

,

with for , where is the length of the string, is the gravitational acceleration, and is the zero-order Bessel function of the first kind; the are the eigenfunctions. Only one term of the expansion, for a single value of , is plotted. The constants and are chosen here to be 1 and 0, respectively.

Reference

[1] P. Hagedorn and A. DasGupta, Vibrations and Waves in Continuous Mechanical Systems, Chichester, England: John Wiley & Sons, 2007.